## Abstract

A vector ray tracing (VRT) model is developed to simulate optical caustic structures in the primary rainbow region of light scattering from oblate droplets. The changes of the optical caustic structures in response to shape deformation of oblate droplets are investigated. Then the curvature calculated from the simulated rainbow fringes is compared with that from the measured rainbow fringes and good agreement is found. Furthermore, according to the generalized rainbow patterns and the relation between aspect ratio and curvature of the rainbow fringe, non-sphericities in terms of aspect ratio of an oblate water droplet is measured with high measurement accuracy.

© 2013 Optical Society of America

## 1. Introduction

Oblate droplets deformed from being spherical under aerodynamic or gravity forces exist in many industrial processes, e.g. fuel injection for combustion, spray cooling, and spray coating. Understanding their light scattering properties is an important basis for developing optical diagnostic techniques to characterize deformed droplets in terms of refractive index, size, shape, and orientation, as well as for evaluating errors arising from the spherical shape assumption by many scattering-based measurement techniques. According to a recent review in [1], optical particle characterization techniques are classified as direct imaging, intensity or intensity ratio, interferometry, time shift, pulse delay and Raman scattering. Interferometric techniques such as phase Doppler [2] and interferometric particle imaging [3] offer perhaps the most accurate determination of droplet size; however both techniques are very sensitive to droplet non-sphericity. Although alternative optical techniques, such as the time shift technique [4], appear to be less sensitive to particle non-sphericity in estimating size [5], it is still necessary to have a quantitative access to the droplet deformation for exact size determination. For characterization of spherical particles in terms of size and temperature, a use of the primary rainbow has been proposed [6]. This technique was generalized to global rainbow thermometry [7, 8], yet still on the assumption that non-spherical droplets provide a uniform background; hence do not influence the interference pattern. Obviously, none of the above methods are readily available for non-sphericity measurements of droplet. On the basis of our previous study of measuring the refractive index and equatorial diameter of an oblate droplet from the generalized rainbow patterns (GRP) [9], we study in this paper the possibility of using optical caustic structures (rainbow fringes) to measure the droplet oblateness - an important indicator of non-sphericity.

Optical caustics exist in light scattering from spherical and oblate droplets. The hyperbolic umbilic (HU) diffraction catastrophe in the primary rainbow region of an oblate water droplet was first observed by Marston and Trinh [10]. The optical caustic structures including the location of cusp caustic and opening rate of the cusp diffraction etc., were successfully described by some approximate theoretical models [11–18]. Most notably, Nye [18] studied the landmark features of the far-field caustics including HU foci, lip events and E_{6} catastrophe according to geometrical optics. In addition, the optical caustics have also been observed from the light scattering by oblate droplets with white light illumination [19, 20]. Both the primary and higher order rainbow caustics of the scattering of light from oblate water droplets [21–23] and internal caustic structures of illuminated liquid droplets [24] have been observed and analyzed. Recently, Debye series has been developed to analyze the formation of rainbow caustic, transverse cusp and HU caustics for a spheroid [25]. Compared to the analysis within the framework of diffraction catastrophe and catastrophe optics [26, 27], Debye series is an exact analysis tool for studying Mie scattering by a spheroid. However, it has not been applied to droplet size of hundreds of microns due to the numerical problems in computation.

In the present study, we developed a vector ray tracing (VRT) model to simulate the optical caustic structures (in terms of rainbow and HU fringes) for an oblate droplet in the primary rainbow region. Since VRT is based on geometrical optics, it pertains to the droplet size over tens of microns used in our experiments. An analytical expression for the location of cusp caustic for the primary rainbow is derived based on geometric optics, which is identical to that derived from Herzberger’s formalism [12]. Then the locations of cusp caustics calculated from VRT are compared with that from the analytic solution and good agreement is found, representing a validation of VRT. The evolution of the optical caustic structures in response to shape deformation of the oblate water droplet is discussed. With the purpose of developing optical techniques for deformed droplet measurement in two-phase flow, we repeated the experiments [10, 12, 14, 15] to measure the generalized rainbow patterns from acoustically levitated oblate water droplets and then compared the observed curvatures of the primary rainbow fringes with the VRT simulations. A relation between the rainbow fringe curvature and the droplet shape is established. With such a relation, the non-sphericity (aspect ratio) of oblate water droplets can be measured with high accuracy.

## 2. Vector ray tracing model

Setting the center of an oblate spheroidal droplet to be the origin of the Cartesian coordinate system, the droplet surface is described by *x*^{2}/*a*^{2} + *y*^{2}/*a*^{2} + *z*^{2}/c^{2} = 1, where *a* and *c* are radii in equatorial plane and along symmetry axis respectively. We then model the optical caustics generated when a light beam illuminates the droplet from its side (namely along the positive y-axis in Fig. 1). The direction of an incident light ray is defined as a unit vector, namely **L**_{0} = (0,1,0). It encounters the droplet surface at the point *A* (*x*_{0}, *y*_{0}, *z*_{0}). Using Snell’s law in vector form, we obtain the direction of refraction ray **L**_{01}:

*m*is the refractive index of the droplet and

**n**

*the surface normal at point*

_{A}*A*. The reflection ray at point

*B*and refraction ray at point

*C*are given by

**n**

*and*

_{B}**n**

*are surface normals at points*

_{C}*B*and

*C*respectively.

Using Eqs. (1)-(2), the direction of the second-order refracted ray associated with the primary rainbow can be calculated in 3D geometry, yielding **L**_{2} = (*a _{x}*,

*a*, a

_{y}_{z}) (see Fig. 2). Projecting

**L**

_{2}onto the

*xy*-plane yields

**L**

_{2}

^{’}= (

*a*,

_{x}*a*, 0). Furthermore, we define

_{y}*θ*as an off-axis angle measured from the

*y*-axis to be

*ϕ*as an elevation angle with respect to the

*xy*plane to be

## 3. Location of cusp caustic

These relations are then used to calculate the location of cusp caustic. As is well known, the cusp caustic is associated with the contribution from two equatorial rays and two skew rays [10–12] and the VRT model allows the cusp caustic to be identified. On the other hand, Nye [11] obtained a derivation for calculating the droplet aspect ratio when the hyperbolic umbilici catastrophe can be observed. Furthermore, an analytical solution to predict the location of cusp caustic was given based on Herzberger’s formalism [12]. In the present study the analytical solution for calculating the cusp location is derived based on geometric optics. In order to compare with Nye’s derivation [11], we use an O-*x*^{’}*y*^{’}*z*^{’} coordinate system shown in Fig. 3. The spheroidal surface satisfies *x*’^{2}/*a*^{2} + *y*’^{2}/*c*^{2} + *z*’^{2}/*a*^{2} = 1. Associated with the incident direction **L**_{0}, the Descartes rainbow ray impacts the droplet at the point *Q* (0, 0, *a*) in the equatorial plane of the droplet. After being refracted into the droplet it intersects the surface at *R* (*a*sin2*θ _{r}*, 0,

*a*cos2

*θ*), where

_{r}*θ*is the angle of refraction at the entry point

_{r}*Q*. Letting the

*Q*shift by an infinitesimal amount

*ε*

_{1}and

*ε*

_{2}in

*x’*and

*y’*directions respectively and keeping the incident direction fixed, the entry point moves to

*Q*(

_{1}*ε*

_{1},

*ε*

_{2},

*-a*(1−

*ε*

_{1}

^{2}/

*a*

^{2}−

*ε*

_{2}

^{2}/

*c*

^{2})

^{1/2}), which is off the equatorial plane. Generally, the refraction of the skew ray passes above or below the point

*R*. If the skew ray also passes

*R*, the exiting ray will remain parallel to the equatorial plane to first order according to the symmetry of the oblate spheroid. It means that the skew rays focus in the vertical direction as well as that in the horizontal plane.

The direction of the incident ray in the O-x^{’}y^{’}z^{’} coordinate system is given by **L** = (sin*θ*_{i}, 0, cos*θ*_{i}), where *θ*_{i} is the angle of incidence at point *Q*. To the first order, the normal to the droplet surface at *Q*_{1} is **n**_{1} = (*ε*_{1}/*a*^{2}, *ε*_{2}/*c*^{2}, −1/*a*). Provided that the skew ray passes through *R*, the ray of refraction is **L ^{’}** = (

*a*sin2

*θ*

_{r}-

*ε*

_{1}, −

*ε*

_{2},

*a*cos2

*θ*

_{r}+

*a*). The ray of incident direction

**L**, the normal

**n**

_{1}at the incident point and the refracted ray direction

**L**should be coplanar. It means that the vector triple product is zero ($L\times {n}_{1}\times {L}^{\text{'}}\equiv 0$), namely:

^{’}Equation (7) is as same as that obtained by Marston using Herzberger’s formalism [12]. For the spheroid with an aspect ratio satisfying the condition Eq. (7), the skew rays will focus vertically. The two skew rays are above and below the equatorial plane respectively. Together with the two equatorial rays, they focus in the same direction, giving rise to the cusp caustic. According to geometrical optics, the scattering angle of the cusp ray is given by *θ* = π + 2*θ*_{i}−4sin^{−1}(sin *θ*_{i}/*m*) and the primary Descartes ray satisfies sin*θ*_{i} = (4−*m*^{2}/3)^{1/2}. Substitution into Eq. (7) yields the critical aspect ratio *a*/*c* = [3*m*^{2}/(4*m*^{2}−4)]^{1/2}, i.e. the ratio at which HUFS arises. This result is identical to that given by Nye [11,18].

Then the location of cusp caustic is calculated by use of the analytical solution Eq. (7) and compared with that calculated by VRT simulations. Figure 4 displays such a comparison. The cusp caustic first appears at θ = 165.57° for oblate water droplets with aspect ratio *a*/*c* = 1.069389. When the ratio increases, the cusp moves to smaller scattering angles until it merges with the primary rainbow caustic at θ = 137.92° for *a*/*c* = 1.309779. Then the cusp shifts to larger scattering angles and disappears at θ = 179.96° for *a*/*c* = 1.414742. The agreement between the analytic solution [12] and the VRT simulation is excellent, with only minor deviation due to the finite grid resolution of incident rays.

## 4. Optical caustic structures of the primary rainbow

On the basis of VRT validation in terms of the location of cusp caustic, we then compute the shape of the rainbow and HU fringes, referred to as structures of optical caustics. For the second-order refracted rays (*p* = 2 in geometrical optics), there is a revolution (turning point) at the deflection angle of the primary Descartes ray. This angle is where the primary rainbow fringe forms. In VRT, the rainbow fringe can be identified from infinitively large density of the emergent rays. As the HU caustics are associated with the contribution from skew rays [10–12], it can be identified by the VRT model. Considering the beam divergence in the droplet, a sufficient number of rays have to be used to ensure the numerical accuracy. To display the extremely high sensitivity of optical caustics on aspect ratio of oblate droplet, the precision of the aspect ratio is given to the sixth decimals in some places of the following studies.

For water droplets with the relative refractive index *m* = 1.333 and equatorial radius 100*μm*, but with different aspect ratios, the rainbow fringe and HU fringe in the primary rainbow region are computed. To fully display the caustic structures, only part of the caustic structure (within −10°≤*φ*≤10°) is given in our numerical demonstrations. Figure 5(a) shows the primary rainbow fringe for a spherical droplet (*a*/*c* = 1). For this aspect ratio the primary rainbow fringe exhibits a weak curvature with respect to the elevation angles and bends towards larger scattering angles (the backward direction). The left-most point (apex point) of this fringe is the rainbow caustic point in the equatorial plane. Its relevant pattern is the fold diffraction catastrophe symbolized by *A*_{2} in catastrophe optics [27]. The cusp caustic first appears for a droplet with aspect ratio *a*/*c* = 1.07 (θ = 165.52°), which is consistent with theoretical prediction [12,18]. Four rays are responsible for its formation. Two are in the equatorial plane focusing horizontally whereas the other two are skew rays offset vertically from the equatorial plane but also focusing in the horizontal direction. For *a*/*c* = 1.07, only a cusp point appears in Fig. 5(b). When the aspect ratio further increases, the HU fringe unfolds in Fig. 5(c) and the cusp caustic shifts towards smaller scattering angles (the forward direction). Further increment of the ratio makes the HU fringe gradually unfold, as shown in Fig. 5(d). Together with the unfolding of HU fringe, the rainbow fringe exhibits an increased curvature because the primary Descartes rays off the equatorial plane shift backward more as the aspect ratio increases. For a droplet with *a*/*c* = 1.27, the rainbow fringe partly overlaps with the HU fringe (Fig. 5(e)) and for *a*/*c* = 1.31 the cusp caustic merges completely with the primary rainbow caustic (Fig. 5(f)), creating the so-called hyperbolic umbilic focal section (HUFS) symbolized by${D}_{4}^{+}$, which is consistent with experiment [10] and theoretical prediction [12,18]. HUFS is described by three control parameters *C*_{1}, *C*_{2} and *C*_{3} in catastrophe optics [27] and its apex angle is experimentally measured to be 43.5 ± 1° [10]. In VRT simulation the apex angle is 43.42°, which agrees very well with the experimental value, exhibiting only slight differences. An analytic expression for calculating the apex angle as a function of refractive index was given by Marston [16], predicting it as 42.1°. The expression was also given in [19]. The main reason for this small deviation of the apex angle between VRT simulation and the analytic expression is due to the fact that the two arms of the rainbow fringe are highly curved at the apex and the mesh size in the VRT simulation is not infinitely small. When the ratio further increases, the cusp shifts further backward and the HU fringe gradually degenerates (Fig. 5(g)). For the aspect ratio *a*/*c* = 1.414742 (Fig. 5(h)), the fringe collapses into the cusp caustic point at 179.96° also agreeing with theoretical prediction [12,18]. Then no cusp caustic is observed for 1.414742<*a*/*c*<1.525000 (Fig. 5(i)). However, it reappears at large ratios such as *a*/*c* = 1.60 (Fig. 5(j)), creating the *E*_{6} diffraction catastrophe also called symbolic umbilic focal section [27]. Note that for *E*_{6}, the rainbow fringe is bent towards smaller scattering angles. The progression of optical caustic structures in response to the change of aspect ratio of oblate droplet is consistent with change of the first and HU bows of the generalized rainbow patterns observed in experiment [10] and Nye’s simulation based on geometrical optics [18].

Then Fig. 6 displays the primary rainbow fringes for oblate droplets with the same refractive index (*m* = 1.333) and aspect ratio (*a*/*c* = 1.21) but different equatorial radii (100µm, 500µm, and 1000µm respectively). All rainbow fringes present the same concave shape (bending towards larger scattering angles), symmetric about horizontal plane and overlapping completely due to the independence of rainbow fringe on droplet size. Attention has to be paid to the fact that, the interference fringe is dependent on the droplet size. And the “Airy shift” between the angle of the main rainbow peak and the Descartes rainbow angle varies as the droplet size changes. In this work, the rainbow fringe referred to the Descartes rainbow angle is independent on the droplet size. However simulations for other type of liquid droplets show that the rainbow fringe shape is dependent on the refractive index of droplet. And then, the comparison of the rainbow fringes for oblate droplets with same aspect ratio and equatorial radius but different refractive indices are shown in Fig. 7. The rainbow fringe for silicone oil droplet (*m* = 1.400) is bent much more than that for water droplet (*m* = 1.333). The rainbow caustic appears at 146.75° for a silicone oil droplet in the equatorial plane while at 137.92° for the water droplet.

In addition, Fig. 8 shows the primary rainbow fringes for oblate water droplets with aspect ratios 1.00, 1.10, 1.21 and 1.25 respectively. As the aspect ratio increases, the rainbow fringe exhibits an increase of curvature at the apex point and a decrease of the opening rate of the rainbow fringe. For all these aspect ratios, a prominent feature is that the location of the rainbow caustic point in the equatorial plane remains unchanged. This is because the cross-section of the oblate droplet remains circular in the equatorial plane so that the second-order refracted rays always exit at the same angular direction in the equatorial plane of the droplet. Further increasing the aspect ratio brings in the rainbow fringe, firstly unfolding and then folding and the appearance of HU fringe as shown in Fig. 5. It is demonstrated from the VRT simulations that the rainbow fringe is dependent on the refractive index and aspect ratio but independent on the equatorial radius of an oblate droplet.

## 5. Non-sphericity measurement of oblate water droplets

To observe the generalized rainbow pattern and optical caustics, an ultrasonic acoustic levitator is used to suspend a water droplet and deform it to different aspect ratios through adjusting the acoustic pressures. For rainbow pattern measurement, the droplet is illuminated by a linearly polarized He-Ne laser beam with a wavelength of 632.8 *nm*. A CCD camera is positioned in the primary rainbow region, i.e. around 138* ^{o}* for a water droplet, and is focused at infinity so that the recorded light scattering pattern is equivalent to that observed in the far field [10]. The viewed region of the camera is 137.3°<

*θ*<143.5°, −2.3°<

*ϕ*<2.3°. To directly observe the droplet shape for later comparison, the droplet image is recorded by another CCD camera with backlighting. Figure 9(a) shows the generalized rainbow pattern for a nearly spherical water droplet with radius

*a*= 0.80

*mm*and aspect ratio

*a*/

*c*= 1.03. Notably, the bows of the pattern in Fig. 9(a) are almost straight lines vertical to the horizontal plane. And the profile of the bows in Fig. 9(a) are similar to the primary rainbow fringe for

*a*/

*c*= 1.00 shown in Fig. 8. Figure 9(b) displays the pattern for a water droplet with

*a*= 0.84

*mm*and

*a*/

*c*= 1.23. Compared with Fig. 9(a), the bows in Fig. 9(b) become noticeably bent and the cusp caustic appears at the right of pattern. The evolution process of the bows’ shape as the result of the water droplet deformation reveals the possibility of measuring the oblateness of droplets from the curvature of the corresponding rainbow fringes.

Then a comparison of the curvature of the rainbow fringes obtained from experiments and that from the VRT simulations is shown in Fig. 10. As is the convention for comparison, only the curvature at the apex point of the rainbow fringe is calculated. Note that due to the droplet vibration, the rainbow fringes obtained experimentally are not as smooth as the theoretical shapes shown in Fig. 8. Therefore a regression fitting was used to achieve a smoothing. The curvatures calculated from experiment are shown with error bars corresponding to one standard deviation (68% confidence level). It can be seen that the agreement between the VRT model and measurements is excellent at most aspect ratios: most deviations are within the experimental uncertainty. Although not shown here, similar results have also been obtained for silicone oil droplets with refractive index *m* = 1.400. Attention has to be paid to the fact that, for large aspect ratios (e.g. a/c>1.23), the droplet vibrates significantly in the levitator and the generalized rainbow pattern becomes highly instable and blurry. So it is difficult to distinguish the rainbow fringe without using a high-speed camera. For this reason the curvature of the rainbow fringes are shown only for the droplets with aspect ratios *a*/*c*≤1.23.

Furthermore, based on the relation between the rainbow fringe curvature and the aspect ratio of oblate water droplets (Fig. 10), the aspect ratios (*r*_{GRP}) of droplets are inverted from the corresponding generalized rainbow patterns and compared to observed values (*r*_{Imaging}) from the directly recorded droplet images. Figure 11 gives such a comparison. The inverted aspect ratios are also indicated with error bars corresponding to one standard deviation. It can be seen that the measured aspect ratios agree well with that from direct imaging. To evaluate the accuracy of inverted *r*_{GRP}, the relative error in percentage (∆ = (*r*_{GRP} − *r*_{Imaging})x100/ *r*_{Imaging}) is given in Fig. 12. The relative error lies between −1% and 1% - indicating high accuracy of droplet non-sphericity measurement from the corresponding generalized rainbow patterns.

## 6. Summary

By developing the VRT model, optical caustic structures including the rainbow and HU fringes in the primary rainbow region of light scattering from oblate water droplets are simulated. As a validation of VRT model, the location of cusp caustics are calculated by VRT and compared with that by analytic solution, in which a good agreement is found. Then the evolution of the optical caustic structures in response to shape deformation of oblate water droplets is investigated by VRT simulation and is found to be consistent with experimental observation very well. For a given type of droplet, the curvature of its rainbow fringe is a function of aspect ratio of the oblate droplet. Therefore the non-sphericity (aspect ratio) of an oblate droplet can be inferred. With the relation between aspect ratios and curvatures of rainbow fringes, the aspect ratios of oblate water droplets are measured, with the relative errors less than ~1%. Together with measurement of refractive index and the equatorial radius of the oblate water droplet inverted from the position of rainbow and angular frequency of the generalized rainbow pattern [9], we obtain more complete set of droplet properties including shape, size and refractive index.

## Acknowledgment

This research is supported by the German Research Foundation under Grant No. TR 194/49-1. H. Yu is also grateful to the Graduate School of Computational Engineering and the Research Training Group GRK 1114 at the Technische Universität Darmstadt for supporting his Ph.D. program.

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